Voronoi Summation Formulae on GL(n)

TL;DR

This paper introduces new Voronoi summation formulae for automorphic forms on GL(n) with n≥4, expanding the toolkit for analyzing automorphic representations and their Fourier coefficients.

Contribution

The paper discovers multiple new Voronoi formulae for GL(n), generalizing previous results and involving hyper-Kloosterman sums for n≥4.

Findings

Identifies [n/2] new Voronoi formulae for GL(n)

Formulates these as Poisson summation formulae with Fourier coefficients

Involves twists by hyper-Kloosterman sums

Abstract

We discover new Voronoi formulae for automorphic forms on GL($n$) for $n\geq 4$. There are $[n/2]$ different Voronoi formulae on GL($n$), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.